Mutual Z in a square array

The "four-square" array of vertical antennas has been around a long
while in various forms. The idea is to put 4 vertical antennas in a square with
sides that are 1/4 wavelength, 3/8 wavelength, etc.. The array is symmetrical,
and a fairly simple feed scheme (feed 2 in phase, and 2 lagging by 90 or 135
degrees) and some relays can be used to form a beam that can be rotated to 4
(or 8) positions. Given that the beamwidth is on the order of 90 degrees, this
works fairly well. The moderately large aperture also means that not too much
power is being radiated at high angles.

This page gives some mutual impedances calculated for an array in free space
using NEC2.

Historical notes

It seems to be quite popular for 160m, where the reasonable aperture with small
radiators, and the ability to point away from noise is probably the selling
point. The ARRL handbook had a 40 meter version using a Wilkinson power divider
for a number of years. Unfortunately, that design didn't allow for the mutual
Z of the elements, and didn't perform particularly well, a topic discussed at
some length in the ARRL Antenna Book. Subsequently, the Antenna Compendium has
had a number of versions, including some with center loaded sleeve dipoles,
various schemes using sloping dipoles off a tower, and so forth.

I've often thought about building a small monopole array for use at 2m and
70cm by putting 4 antennas on the roof of the car, and using a suitable rotary
switch to move the beam. The extra 5-6 dB or so from the array gain would help
hit those distant repeaters. On the other hand, a 40 Watt amplifier might work
just as well, and is omnidirectional.

Modelling results

Some NEC studies were done to determine the mutual Z for a square array. Two
cases were run with the square sides 1/4 wavelength and 3/8 wavelength. (At
1/2 wavelength, you don't form a unidirectional beam). In both cases, the length
of the dipoles were adjusted to make the driven element "resonant",
or at least, with a reactive impedance less than an ohm.

The 4 elements are numbered as follows:

2

4

1

3

There is a lot of symmetry: Z12=Z13, Z14 = Z23, etc..

The model was run at 29.98 MHz with copper wires 0.00127 in diameter (approximately
#10 AWG). The wavelength of 10.0 meters made spacing the elements easy, which
is why it was chosen. Element 1 was driven, and the other elements were loaded
with a 10K resistor. The voltages and currents from the NEC output were then
used to calculate the mutual Z.

In the 1/4 wave case, the elements wound up being 4.83 meters long. In the
3/8 wavelength spacing case, the elements were slightly shorter: 4.82 meters.
The one cm difference manifested itself as a change in the reactive impedance
of about 1 ohm. While the impedances in the following table are reported to
0.01 ohm, I suspect that the real accuracy is somwhat poorer.

Spacing
(wavelength)

Z11

Z12

Z13

Z14

1/4

69.77-j0.87

-36.16+j29.76

-36.16+j29.76

-11.35+j37.99

3/8

67.64+j0.82

-9.98+j35.16

-9.98+j35.16

17.41+j22.45

Note that the "resonant" impedance is slightly different from the
nominal 72 ohms you would expect for an ideal thin dipole. This is no doubt
due to several factors: the finite size of the wires, the resistive loading,
and the effect of the "parasitic" elements, even though they are effectively
open circuited.